Isomorphic Binary Structures

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Formal Definition

Let $\langle S,*\rangle$ and $\langle S',*'\rangle$ be binary algebraic structures. An isomorphism of S with S' is a one-to-one function $\phi$ mapping S onto S' such that $\phi(x*y)=\phi(x)*'\phi(y)$ for all $x,y\in S$.If such a map $\phi$ exists, then S and S' are isomorphic binary structures, which we denote by $S\simeq S'$, omitting the $*$ and $*'$ from the notation.
Informal Definition

$S$ and $S'$ are isomorphic binary structures if $S$ and $S'$ are isomorphic. $S$ and $S'$ are isomorphic if there is a mapping that preserves sets and relations among elements.


Let $2\mathbb{Z}=\{2n|n\in\mathbb{Z}\}$, so that $2\mathbb{Z}$ is the set of all even integers, positive, negative, and zero. We claim that $\langle\mathbb{Z},+\rangle$ is isomorphic to $\langle\mathbb{2Z},+\rangle$.
Step1 The obvious function $\phi:\mathbb{Z}\rightarrow2\mathbb{Z}$ to try is given by $\phi(n)=2n$ for $n\in\mathbb{Z}$.
Step2 If $\phi(m)=\phi(n)$, then $2m=2n$ so $m=n$. Thus $\phi$ is one to one.
Step3 If $n\in\mathbb{2Z}$, then n is even so $n=2m$ for $m=n/2\in\mathbb{Z}$. Hence $\phi(m)=2(n/2)=n$ so $\phi$ is onto $2\mathbb{Z}$.
Step4 Let $m,n\in\mathbb{Z}$.The equation$\phi(m+n)=2(m+n)=2m+2n=\phi(m)+\phi(n)$
then shows that $\phi$ is an isomorphism.


The binary structure $\langle\mathbb{Q},+\rangle$ and $\langle\mathbb{R},+\rangle$ are not isomorphic because $\mathbb{Q}$ has cardinality $\aleph_0$ while $|\mathbb{R}|\neq\aleph_0$.

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